First, let me ask............does this LT1000 have an electric PTO clutch or do you engage the deck by manually moving a lever?
The reason this is pertinent is due electric PTO clutches requiring a higher amp output alternator to satisfy the load of the PTO clutch AND recharge the battery while you are mowing.
If you engage your deck manually your LT1000 likely has the low amp output alternator. If this is the case, it takes about 30 minutes (average) run time at high RPM just to completely recharge the battery from the drain of starting when you begin mowing.
So.......if you have the manual PTO and have not been running the engine a long time between starts during your hauling and carting business.......you have been drawing more out of the battery than you have been putting back in.
But, if your LT1000 has an electric PTO, it would perform better at recharging under short run time duty because you are not using the PTO while towing.
I would guess that even that much extra output could be insufficient if you do a bunch of starts and then run it for less than 5 minutes after each crank.
An easy way to check whether the alternator is putting out to spec is to check the voltage of the battery (with the engine not running) AFTER it has been fully charged.
The battery should read about 13.4 to 13.8 volts.
Next, crank the engine and run it at high RPM and read the battery voltage.
It should read AT LEAST the same voltage as with the engine off and maybe slightly higher.
Allow the engine to run at high RPM for at least 15 minutes and then read the battery voltage while the engine is still running. If the alternator is performing well the voltage should have risen to about 14.2 volts as the battery was recharged from the drain imposed on it in cranking the engine earlier.

The easiest way to check if the charging system is working is to start the tractor and turn on the headlights.

Watch the headlights with the tractor idling.

Run the throttle to wide open... do the headlights get brighter?

If yes, then the charging system is working.

Remove the battery and take it to an auto parts store that will load test it for you and right in front of you.

Thanks for the replies. It is a manual PTO, so I would guess that the short cycle times are the problem. So from now on, will put it "turtle" mode :) and engage parking brake instead of turning off/on. Pretty sure that will fix the problem. If not, I'll test the charging system as instructed. As usual, great site, thanks for the help.

While the headlight test will provide evidence of a functioning alternator in many cases.........it is not proof in ALL cases.
Briggs build lots of engines configured with the "Dual circuit" alternator.
The Dual circuit system has 2 separate outputs.
1 output is unregulated DC power to charge the battery.
The other output is unregulated AC power that operates only the headlights.
To tell whether a Briggs has the Dual circuit system you just turn on the headlight switch with the engine stopped.
If the lights burn when the engine is not running, the charging system IS NOT a Dual circuit type.
If the headlights only burn with the engine running, it has a Dual circuit system.
I call the Dual circuit alternator a minimalist system.

In most instances with contemporary L&G products the headlight test will indicate charging without the need for tools and meters.

Mownie is correct. I did say "the easiest way..." not the most technically correct way.

It is always best to start at square one with charging problems especially since Harbor freight sells a nifty DVM for about $3. Everyone should have one (or three) and learn to use it.

Typical charging output for an LT 1000 is 3 amps DC, with 5 amps (AC) for the headlight circuit.